216 CHEMICAL ENGINEERING Crushing Surface Diagram.—In view of the importance of crushing efficiencies in various industries, and the frequent discussion taking place on this subject, the diagram which may be called the crushing-surface diagram is submitted as a means of comparison and study of crushing conditions. The scheme was discovered during the summer of 1910, when trying to understand why one tube mill did not do the work expected of it on the basis of what another was doing on " a similar ore just as hard." First it should be stated that this diagram is based on Rittinger's law, which is interpreted to mean that the work done in crushing is proportional to the surface exposed by the operation, or better expressed for this purpose, the work done on a given mass of rock is proportional to the reciprocal of the diameter of the final product, assuming that all the mass has been reduced to one exact size, which is only theoretically possible. Kick's law is frequently referred to in connection with this subject, especially since the publication of H. Stadler's1 work, and is thus expressed: "The energy required for producing analogous changes of configuration of geometrically similar bodies of equal technological state varies as the volumes or weights of these bodies.'7 This law does not apply so much to crushing as to deformation of bodies before rupture takes place. In Fig. 12 are represented two particles of ore "of equal technological state" shown as cubes between the faces of a crushing or testing machine. Assuming the theoretical mesh, equivalent to the reciprocal of the diameter, and using concrete values, we have a 100-mesh particle with eight times the volume of the 200-mesh particle, and with an area per face four times that of the 100-mesh particle. The dimensions are as 2 to 1, and the bodies being similarly deformed within the elastic / \ 1 V 1 1 \ / 1 i FIG. 12.—Kick's law. 100-mesh cube 200-mesh cube Area one face = A Energy = PAD Area one face — — .'. Energy proportional to volume. limit without fracture, the energy that must be applied in each case to produce this deformation is the product of the average resisting force per square inch, the same in both cases, by the area worked against and by the distance through which this average force works. As shown in the figure, in this particular instance the energy absorbed is proportional to volume and it can be similarly shown for the general case. On the gradual release of the external pressure the energy absorbed is given back to the machine producing the deformation and the body returns to its original shape. It should be noted that the body has been deformed only by a gradually increasing pressure, the first increment of deformation not requiring so much pressure as the last. In case the body has been deformed beyond its elastic limit either the whole mass of particles has been reduced to the molecular state by the freeing of their bonds with 1 Trans. I. M. M., Vol. 19, "Grading Analyses and Their Application."